1. **Problem Statement:** We have temperature data for City X and City Y over 30 days. We need to calculate the range, variance, and standard deviation for each city, determine which city has a more stable climate, and analyze the effect of increasing City X's temperatures by 5°C on its standard deviation. 2. **Formulas and Important Rules:** - Range = Maximum value - Minimum value - Variance $\sigma^2 = \frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^2$ where $\bar{x}$ is the mean - Standard deviation $\sigma = \sqrt{\text{variance}}$ - Adding a constant to all data points shifts the mean but does not change variance or standard deviation. 3. **Calculations for City X:** - Data: 25, 27, 26, 28, 30, 29, 31, 32, 33, 35, 36, 34, 33, 31, 30, 29, 28, 27, 26, 25, 24, 22, 21, 20, 19, 18, 17, 16, 15, 14 - Range: $36 - 14 = 22$ - Mean $\bar{x} = \frac{\sum x_i}{30} = \frac{25+27+...+14}{30} = 25.1$ - Calculate squared deviations and variance: $$\sum (x_i - 25.1)^2 = 560.9$$ $$\text{Variance} = \frac{560.9}{30} = 18.7$$ - Standard deviation $= \sqrt{18.7} \approx 4.32$ 4. **Calculations for City Y:** - Data: 10, 12, 15, 14, 16, 18, 19, 21, 23, 25, 27, 30, 28, 29, 30, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17 - Range: $31 - 10 = 21$ - Mean $\bar{y} = \frac{\sum y_i}{30} = 22.3$ - Calculate squared deviations and variance: $$\sum (y_i - 22.3)^2 = 462.1$$ $$\text{Variance} = \frac{462.1}{30} = 15.4$$ - Standard deviation $= \sqrt{15.4} \approx 3.92$ 5. **Comparing Stability:** - City Y has a smaller standard deviation (3.92) than City X (4.32), indicating City Y's temperatures vary less and thus have a more stable climate. 6. **Effect of Increasing City X Temperatures by 5°C:** - Adding a constant shifts all data points by 5 but does not affect variance or standard deviation. - Therefore, City X's standard deviation remains $4.32$ after the increase. **Final answers:** - City X: Range = 22, Variance = 18.7, Standard Deviation = 4.32 - City Y: Range = 21, Variance = 15.4, Standard Deviation = 3.92 - More stable climate: City Y - Standard deviation of City X after +5°C increase: 4.32